Estimating \(\chi (q;s_1,s_2)\)  at a fixed critical level \(x^q\)  at a set of sites \(s_2\)  can allow us to explore the spatial extent of extreme events. Figure REF  (top panels) shows \(\chi (q; s_1,s_2)\)  calculated across a number of locations in northern England, with \(s_1\)  being Lancaster in this instance. We explore two cases where \(x^q\)  is chosen to the \(90\%\)  quantile and the 10-year return level at each site. In particular, we see that the probability surface decays more steeply as \(|s_2-s_1|\)  increases for the more extreme events. We also see that Liverpool, Manchester and Leeds are more likely than not to experience an event on the \(90\%\)  quantile simultaneously to Lancaster; however, this scenario is less likely for an event corresponding to the 10-year return level. Similarly, in Germany, as shown in the bottom panels in Figure REF , the probability of experiencing an extreme windstorm event simultaneously with Berlin decreases as the event becomes more extreme. The spatial extremal dependence is slightly stronger for Berlin than Lancaster, as might be expected given Berlin is more inland on a large land mass. In both regions, there is little evidence of anisotropy in the extremal dependence structure, with perhaps some indication of stronger dependence in the northwest-southwest direction centred at Berlin. The results in Figures REF  and REF  illustrate how the spatial extent of an extreme windstorm event becomes more localised as the magnitude increases. This result implies that in the limit, extreme values at each location tend not to occur simultaneously, which corresponds to the property of asymptotic independence. Models for asymptotic dependence, that is, when \(\chi (q;s_1,s_2) \rightarrow c > 0\)  as \(q \rightarrow 1\)  for \(s_1 \ne s_2\) , lead to extreme values tending to occur simultaneously, are well-established but tend to over-estimate the probability of extreme joint events given that the underlying process is asymptotically independent, i.e., when \(c=0\) . Models that capture asymptotic independence are less well-established; see [1]}, [2]}, [3]} and [4]} for some examples. We have shown that our model captures the property of asymptotic independence over space, while accounting for the complex non-stationarity of the extratropical cyclone system.
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